I've heard that Lord Buddha experienced an awareness or perception even in deep sleep.

Not in dreams as in lucid dreaming, but without movement of mind, just awareness. As it happens, I have heard that certain Buddhist teachers also experience this as well as some yogis.

This seems to be an awareness or consciousness that does not rise and fall like normal waking consciousness.

I think in yoga they call it Turiya or the 4th state. They think it is permanent reality, even though it is not.

As for myself, there have been some experiences in the last few months when the consciousness seemed to "go under" as in sleep, except these times, even with the conscious mind shutting down even then, I felt awake.

My question is, what is this seemingly continuing "background" awareness called in Theravada? Is is a more subtle form of consciousness that only appears not to change throughout the day and night to those without the insight required to do so?

I've heard that Lord Buddha experienced an awareness or perception even in deep sleep.

Not in dreams as in lucid dreaming, but without movement of mind, just awareness. As it happens, I have heard that certain Buddhist teachers also experience this as well as some yogis.

This seems to be an awareness or consciousness that does not rise and fall like normal waking consciousness.

I think in yoga they call it Turiya or the 4th state. They think it is permanent reality, even though it is not.

As far as I know they think "the Fourth" leads to absolute reality.

As for myself, there have been some experiences in the last few months when the consciousness seemed to "go under" as in sleep, except these times, even with the conscious mind shutting down even then, I felt awake.

Could be, could be not. There are several states that can be mistaken for turiya. For example, practitioners of samatha sometimes find themselves in the state of non-perception. The state of non-perception can be useful for example as a means to stay unaware of pain during a surgery but in itself it does not lead to Liberation.

In yoga Turiya is also called "The Witness". It does not experience wake, dream and deep sleep but observes these states from a detached point of view. It is described as considering wake, dream and deep sleep as alien, far away, and so on. To experience Turiya means that it makes no difference whether there is wake, dream or deep sleep because one is not in those states oneself, aka one is not identified with them. The view is that "something else" is in wake, dream or deep sleep, something alien, remote, far away, something one sees clearly but does not feel to be oneself or having anything to do with oneself (except that it is observed of course).

Currently, I think that "the Fourth" is called vipassana (insight) in Theravada. I mean the state not the technique. It is the state the Buddha was in observing his mind going through the jhanas directly before his Enlightenment. The pleasant feeling of jhana is described as remote and alien, as not entering and not remaining - similar to how Turiya is described.